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Pounds measure weight, my friend. As in the effect of a gravitational field on a certain mass.

Those who use pounds as force use slugs as the unit of mass. Same relationship as mass in kilograms and weight in newtons (i.e. Newton's 2nd Law), except for the weird-ass numbers.

Just how many hogsheads are there in a fortnight, anyway?

...laura

Nope, I'm right. (In fact, I've written a program implementing the Miller-Rabin test myself.)

If you read the Wikipedia article, you'll see:

It can be shown that there always exists a strong witness for any odd composite n, and that at least 3/4 of the values for a are strong witnesses for the compositeness of n.

Also Mathworld says:

If N multiple independent tests are performed on a composite number, then the probability that it passes each test is 1/4^N or less.

And according to this page:

Choose k independent integers a1,...ak which 1 < ar < n - 1. Then if (odd) n passes the Rabin-Miller test for each base ar, the probability that n is composite is no more than 1/4^k.

Find a pretty picture (and math) here

Instead, Tien Kieu from my university wants to solve arbitrary Diophantine equations using quantum effects. If he's a) correct, and b) it becomes possible to create the required quantum behaviours for arbitrary equation, the following problems become solvable:

• The halting problem for arbitrary Turing machines, with all that that would imply.
• The Riemann hypothesis.
• Goldbach's conjecture

  Needless to say, to say people are sceptical of Kieu's ideas is an understatement, but it's fun to speculate about the "what if"...

The Mathworld web site was similar in that copyright was not owned by one person. Possibly the volunteers signed their copyright over to Eric, but on the surface it appears that Eric did not have the authority to (unintentionally) give copyright to CRC. That didn't stop CRC.

It's ambiguous. The current permissions form merely grants Wolfram the right to do whatever they please with the material. I have no idea what the previous terms were.

Wikipedia's content is explicitly under a license that allows for irrevocable rights to republication and derivative works. The worst that can happen without declaring the GFDL invalid is that the Foundation goes bankrupt. But we'd still have the right to take the database and go elsewhere.

Please tread carefully when dealing with Google. I know they claim to 'do no evil', but the Mathworld Mess highlights what can go wrong. Wikipedia is too precious to lose.

1. The studio creates Y = f(x). You want to get Y (it's the best you can do).
2. You play it back to a .wav file to get Z = g(Y) = g(f(X))
3. You know Y. You know g(x). You create h(x) such that h(Z) = Y. This is the inverse function of g(x).

Your trying to do an apples to oranges comparison. A Watt is a unit of power. The bomb is usually measured in kilotons or megatons, energy units. But I'll try:

P = E/t
A Watt is a Joule per second.

The energy released by a ton of TNT exploding as a unit of energy is 4.2 x 10^9 Joules. So divide 10^39 (10,000 trillion trillion trillion watts) by 4.2x10^9 and get 2.38095238 x 10^29 tons of TNT per second. That's 2.38 x 10^26 kilotons or 2.38 x 10^23 megatons.

The Hiroshima bomb released about 12.5 kilotons of TNT of energy. That means at one point the magnetar was releasing the energy at the rate of about 1.9 x 10^25 (19,000,000,000,000,000,000,000,000)Hiroshima bombs per second.

Of course assuming all my math is correct...

Yep, it's called the Pigeonhole Principle, and is tought in first year discrete mathematics.

Cheers,

Michael

Wrong theory. That is for primes in general. "These numbers" refers to just the Mersenne primes.

A Mersenne number is all ones when written in binary. If its prime, it is a Mersenne prime.

A fun sidebar to this is perfect numbers, numbers who factors (other than one and themselves) add up to the number. Ancient Greek numerology, y'know.

It's not at all difficult to show that a Mersenne Prime times the next smaller power of two is perfect. Perfect numbers thus have an interesting binary representation.

Nobody has yet proven that all perfect numbers are of this form, or even that they're all even.

...laura

You can say even more. If M can be written as 2^n - 1, then M is said to be a Mersenne number. If M is also prime, then it is a Mersenne prime. For 2^n - 1 to be a Mersenne prime, n must be a prime number, since we have

2^(ab) - 1 = (2^a-1)(2^(a(b-1)) + 2^(a(b-2)) + ... + 2^(a))

For instance, 2^6-1 is (in binary) 111111 = 1001 * 111, as predicted by the above factorisation.

If p is a prime number and if 2^p-1 is a Mersenn prime, then, as was pointed out above, 2^(p-1)(2^p-1), is a perfect number. Moreover, if N is an even perfect number, then N can be written (uniquely) as 2^(p-1)(2^p-1) where p is a prime number and 2^p-1 is a Mersenne prime.

Wikipedia has a reasonably intelligible introduction to perfect numbers, and MathWorld contains a proof of why every even perfect number must have the form claimed above.

To see why M = 2^(p-1)(2^p-1) is a perfect number when p, 2^p-1 are primes, it suffices to note that s(n), the function that maps an integer to the sum of its divisors (e.g. s(6) = 1 + 2 + 3+6, s(8) = 1 + 2 + 4+8) is multiplicative in the number-theoretic sense, that is to say s(ab) = s(a)s(b) whenever a, b have no prime factors in common. Then evaluating s(M) is simply a case of evaluating it on the factors, which are relatively prime since one is a power of 2, 2^(p-1), and the other is an odd prime, 2^p-1. s(2^p-1) = 2^p-1 + 1 = 2^p (since we have a prime number), and 2^(p-1) = 2^p -1 is an easy formula that is true of all powers of 2. Hence s(M) = 2^p(2^p-1) = 2 ( 2^(p-1) (2^p-1) = 2s(M). That is to say, the sum of all the divisors of M add up to twice M, and if we leave the divisor M itself out of the sum, we see that M is a perfect number.

http://mathworld.wolfram.com/MersennePrime.html
One can explain something to another,
but if said person is unable to relate to the way that one is teaching them then either:
The explainer is a horrible teacher, or, the explainee just simply doesn't understand. And i, for one, am a horrible teacher.
Which is why i say: RTFA

http://mathworld.wolfram.com/MersennePrime.html

No; the pigeon-hole principle is that when you have n slots and >n items, at least one slot must be used more then once. It applies perfectly here.

See http://mathworld.wolfram.com/DirichletsBoxPrincipl e.html.

[The positive integers] are the solution to the simple linear recurrance equation a_n = a_n-1 + 1 with a_1 = 1.

"To generate a true random stream of numbers is incredibly difficult, if not impossible"

Yeah, that was my first thought as well, as computers can only general pseudo-random data.

As I searched a little on the topic, I discovered the following quote from http://mathworld.wolfram.com/RandomNumber.html, which I found interesting:

"It is impossible to produce an arbitrarily long string of random digits and prove it is random. Strangely, it is also very difficult for humans to produce a string of random digits, and computer programs can be written which, on average, actually predict some of the digits humans will write down based on previous ones."

Anyone know of such a program or the theory behind it?

Here's a simple mathematical model to help you understand the effects better.

That's not how it works. LEO is more typically quoted at 180km high anyway; and you'd have to go pretty much that high or higher to get to Tokyo anyway.

No, look. Ignoring the ascent air resistance (which is the same for orbital/suborbital anyway pretty much), as soon as you leave the atmosphere, you are in orbit right? I mean, the orbit may intersect the ground(!), but you are technically in orbit.

Now, to go long distances you need perigee to not be too far below ground level, otherwise you will hit the ground almost immediately.

So perigee has to be close to ground level, and apogee has to be above 100km. So, right away you are describing an orbit that is 98% of the way to being a stable orbit.

I mean, what's a hundred kilometers out of 6200km (the radius of the earth). Right?

If you don't believe me, do the *real* math. As a kick-off check out the vis-visa equation and other equations on that site.

Calculate the delta-v to reach an orbit with apogee at 100km and perigee at 0km. Then add on 2km/s to deal with gravity and atmospheric losses during ascent. That will give you the delta-v for going about 1/4-1/2 of the way around the world.

> Gary Hudson gave it as a rule of thumb on USENET a year or two back IRC.

Well, then he was eiter wrong, or using way too high of an orbit.

Gary Hudson is a leading rocket engineer that has worked on numerous launch vehicles, including an ICBM IRC (which *are* suborbital delivery systems if you bother to think about it). If he says it's 30% reduction in payload; that's what it is. Period.

The (two) diagonals of a 600X600 segment of the 800X600 matrix will not have 600X[sqrt(2)] pixels, but exactly 600 pixels. Each pixel is located in a separate row and in a separate column. In just the same way one cannot use the Pythagorean Theorem (http://mathworld.wolfram.com/PythagoreanTheorem.h tml) to find the pixels on the diagonals of the 800X600 matrix. Instead, those diagonals will effectively have one pixel for each of the 800 columns of the matrix.

This great practically self-taught Indian mathematician might have said differently.

Also, a brief look into the history of mathematics will reveal a decimal system in use in India around 2100BC, the development of theories of a solar-centric solar system, and pi around 500 AD, and tangible proof of the development of zero and negative numbers around 650-ish AD (the 7th century, and yes, this is a huge accomplishment nit-wit). Additonally, the term sine is derived from an Indian word, as trigonometry originated there, though you likely never made it through algebra.

The contributions made by the people of the Indian subcontinent are far from trivial. Sounds like someone also needs a history lesson.

http://scienceworld.wolfram.com/physics/SpecialRel ativity.html

Not a huge help. Basically the equation tells you what % of YOUR (the observer) 'time' (delta t) the person moving at velocity 'v' perceives -- e.g. the equation is used to calculate time dilation.

Same idea: E = mc^2 is incomplete. The full equation involves velocity via:

E = m*(gamma)*c^2* = m*c^2/(sqrt(1-(v/c)^2))

The matrix determinant isn't a matrix -- it's just a scalar.

The Wronksian is a perfectly good matrix and useful.

I don't think you really understand the need for gravitational assists. Without such assists a probe using current launch technology would not be able to make it anywhere close to the edge of the solar system. We can't just fire these things off in a roughly straight line. They're launched in what are essentially variations of a Hohmann transfer orbit. Instead of moving from one circular orbit to another though, you're transfering from a circular orbit to a hyperbolic one. The same concept applies, energy needs to be added to increase the altitude of the orbit which is where the gravity assist comes in.

Shooting a probe off perpendicular to the mean plane of the solar system would be shooting yourself in the foot. You'd be entirely missing out on the enormous gravitation boost Jupiter and/or Saturn offer. You would need überamounts of extraordinarily energetic fuel to make the trip without using the gas giants. Even then the rocket (as mentioned) would need to be about 98% fuel. We can at best build ones that are 90% fuel.

There's also a shitload of vaccuum in the planetary plane just as there is perpendicular to the plane. There would really be little use in avoiding the planetary plane as you suggest. Besides the boost the probe would get from the gas giant(s) it could be put on a course where some secondary science could be conducted on the way to the edge of the solar system. You wouldn't need extra equipment to study doppler shifts as the probe entered say Jupiter's gravity well during an assist or watch it's radio signal as it was occulted by the planet for frequency varations to infer magnetic fields or ionized particles in the atmosphere.

Actually, it's not NP, it's most certainly polynomial. Turns out that this "scheduling problem" can be arranged into a matroid, and therefore a simple greedy algorithm will work. Basically, this will *always* just end up with the next "highest priority" item that fits into the current time slot.

(For a final note, I'm not in any way affiliated with mathworld.wolfram.com, but it is a useful reference.) ;-)

Actually, it's not NP, it's most certainly polynomial. Turns out that this "scheduling problem" can be arranged into a matroid, and therefore a simple greedy algorithm will work. Basically, this will *always* just end up with the next "highest priority" item that fits into the current time slot.

(For a final note, I'm not in any way affiliated with mathworld.wolfram.com, but it is a useful reference.) ;-)

Don't think that's true for pi. It might be true for a normal number. You can find a normal number by picking a random point on a ruler and measuring its location to an infinite accuracy. Good luck on achieving the infinite accuracy bit though.

Far be it from me to step in a pile of God doodoo, especially on /. but...
There is a pattern of sorts in the digits of Pi. Providing you don't mind working in hexadecimal, [why that base and no other?...any math PhD's in the audiance?] formula 29 on this page gives you a way to calculate any arbitrary digit of Pi without running a series calculation up to that digit's precision. If a formula for any digit, with independence from all other digits doesn't stretch the definition too much, I'll call it a pattern in the digits of Pi.
I for one welcome a creator who would leave us such puzzles.

A Java database?

Isn't that a little bit like writing a Fast Fourier Transform in LISP?

There are lots of things that Java is perfectly suited for. Databases are not one of those things.

...only for things so obvious that it's common knowledge... Murky area:

How about the snowman accessory kit (carrot, coal, scarf [5380237 and others]) or the prime number patent?

a tool that proves automatically that a program is correct before allowing it to run

a tool that proves automatically that a program is correct before allowing it to run

I was looking at the math section and glad to find an introduction to unitarity, but I wonder if someone who needs an introduction to complex numbers, polynomials, matrices, etc., is going to be able to absorb quantum mechanics in the same read. However, since my mathematical background only covers the first n, where n < page_count, pages of that chapter, it's probably a good call not to make assumptions and start from the beginning. Besides many of us with mathematical backgrounds, like me, get rusty after years of work which doesn't call for much math.

I heartily agree with other posts that this stuff looks like good wikipaedia fodder, probably across many entries. I also wonder if some of this might be able to contribute to Eric Weisstein's sites. He still has quite a few entries that can be added to.

A limit can be infinity, yes. Wolfram's MathWorld shows that the limit of 1/x as x approaches 0 equals +infinity.

since objects need mass to be affected by gravity

Good God NO! Where have you been sicne 1915? Did you miss General Relativity? Space-time is curved so that when the light follows a straight path near a massive body it curves.

By the way, momentum = Energy/c = h*f/c. p=mv for non-relativistic particles, and guess what? Photons are sure as hell relativistic.

To be complete and give more information that relating to the parent, the full equation (c/o Einstein) is E^2-p^2c^2=m^2c^4. Thus one can see that for m=0 (like a photon) E=pc. If one tries to use p=mc for a photon then you get that E=Sqrt[2]mc^2 oops or if you plug in E=mc^2 to this equation you do NOT get =mv.

More information here.

since objects need mass to be affected by gravity

Good God NO! Where have you been sicne 1915? Did you miss General Relativity? Space-time is curved so that when the light follows a straight path near a massive body it curves.

By the way, momentum = Energy/c = h*f/c. p=mv for non-relativistic particles, and guess what? Photons are sure as hell relativistic.

To be complete and give more information that relating to the parent, the full equation (c/o Einstein) is E^2-p^2c^2=m^2c^4. Thus one can see that for m=0 (like a photon) E=pc. If one tries to use p=mc for a photon then you get that E=Sqrt[2]mc^2 oops or if you plug in E=mc^2 to this equation you do NOT get =mv.

More information here.

Okay, allow me to restate. The information and energy content propagates at c. In most cases, this corresponds to a group velocity of c. In the case of X-waves and certain other weird situations, the peak exhibits interference effects causing it to transitorily appear to move faster than c. No information is transmitted faster than c, and the energy doesn't propagate faster than c.

Normally group velocity *is* the velocity at which energy and information propagates in a wavefront. So this is a bizarre exception resulting from the formal definition of group velocity and anomolous dispersion effects. But there is no superluminal information transmission which (back when I was studying physics) was considered one and the same as superluminal group velocity.

Okay, allow me to restate. The information and energy content propagates at c. In most cases, this corresponds to a group velocity of c. In the case of X-waves and certain other weird situations, the peak exhibits interference effects causing it to transitorily appear to move faster than c. No information is transmitted faster than c, and the energy doesn't propagate faster than c.

Normally group velocity *is* the velocity at which energy and information propagates in a wavefront. So this is a bizarre exception resulting from the formal definition of group velocity and anomolous dispersion effects. But there is no superluminal information transmission which (back when I was studying physics) was considered one and the same as superluminal group velocity.

I beg to differ. Next time, troll on a topic you might actually be somewhat competent in.

On the positive side, if someone ever asks you to name a bad application of the hyperbolic tangent, now you know the answer: as a transition between the title bar and the menu items in a web browser.

I think Steve Martin said "Googolphonic"
Googol being a rather large number.

Some guy has a law that says you need to sample at a rate twice as frequent as the signal your sampling. Makes sense if you think about it.

That "some guy" would be Nyquist, who found that you need to sample at least twice as fast as the highest frequency in a waveform if you want to be able to reconstruct that waveform. Sample at a lower rate and you run into "aliasing".

As a visual example, look at the spinning wheels of a car on a TV show. As the car starts moving, the wheel looks like it's spinning forward, because it's spinning slower than half the sampling rate of the TV cameras. As the wheel increases in rotational speed, it will eventually look like it's rotating in reverse, or even stopped. That's because it's now gone past the Nyquist frequency for reconstructing the proper spinning of the wheel, and there's no longer enough information to accurately depict the wheel. Wheels don't do this in real life, because your eyes sample fast enough to reconstruct the full waveform.

Possibly I didn't express myself clearly enough before. Interaction between waves and matter is very, very well understood. The equation has Planck's constant as a coefficient. Planck's constant is a very small number.

I don't have time now to go into this, but I read the paper you cited. Everything I say on Slashdot is only my opinion. My opinion is that the man who wrote knows that he is lying, and is therefore a liar. If I were dean at that university, I would seriously investigate whether he should be fired, or merely re-trained.

For those mathematically interested fractal compression tries to find a matrix transform (A) so the serie x_n = A * x_{n - 1} converges where x is an image matrix and x_{\infty}=image to compress is a fixed point.

This is all from memory. I spent quite a while researching this all about ten years ago in order to solve an argument about whether a solution to chess was possible or not.

The number is extremely high (I've lost my references to how high, but it was rather ridiculous) if you include the 'position repeated 3 times causes a draw' move, which means that positions where all the pieces are in the same place on the board can actually be one of many different positions, depending on how it was reached.

If you play without this rule, the history you need to keep is very limited (basically, whether each side has moved king or either rook, and which player is next to play), so the number of positions is substantially reduced.

As a good first estimate, there are about 65!/33! ways of placing the pieces on the board (including the option of not placing them on the board; the actual answer is slightly more complex than this, but this is close enough for a first estimate). Multiply by two to consider whose turn it is. This is about 10^67. The actual number of possible positions will be less than this.

I believe the number of positions of major pieces alone (discarding the pawn positions, which are harder to calculate the legality of) is in the order of 10^25. Some of these will be legal in one player's turn, but not the other, so including the turn in the information stored doesn't quite double the figure. There number of legal pawn positions is probably a few orders of magnitude smaller, but some of the piece positions are obviously not legal with some of the pawn positions, so we can't just multiply these figures together.

Given these restrictions, a top-of-the-head guess is about 10^40 different legal positions.

Having just googled for other sources, it seems I'm pretty close. Mathematical papers on the subject have given various values between 10^40 (Beeler et al) and 10^43 (Shannon), but with a few venturing as high as 10^120 (which probably includes relevant game history, not just position on board, by the sounds of it).

Here's a reference.

Its hard to explain in a lot of detail and in the space avaliable here. Unfortunately I am not aware of a good online reference. If your interested in looking into the math this http://scienceworld.wolfram.com/physics/Polarizati on.html provides some detail on polarization.

the indexes' entry for "Recursive Procedure" includes the page number of the entry itself

That index entry looks more like an example of a loop than recursion. Recursive functions should have an end-condition. The entry compounds the confusion between loops and recursion unless it is trying to make the point that any recursive function can also be implemeted as a loop.

Here is the same joke using self-recursion.

What happens, though, is that the wavefront becomes larger , and you have to build antennas also larger to capture the same amount of energy.

But, for example, if you completely surround the transmitter with your antenna then you will reclaim all the transmitted energy. This is one of fundamental principles of the field theory. Mathworld has a very good explanation that leads into Maxwell equations.

Also, the path loss in a given direction does not depend on the antenna.

Personally, I believe in logic. But it's a belief, and it will never be more than that.

It certainly won't ever be more than that, if you never educate yourself out of your woeful misunderstanding of Gödel's theorem.

Other posters have provided some good responses, but there's another angle which I don't think has been mentioned.

Gödel showed that certain kinds of formal systems have certain limits. However, those limits are clearly defined. You still know that if you are able to prove a conclusion within a consistent system, that the proof is correct within that system (and vice versa for disproof).

You seem to be confusing this result of Gödel's with the relationship between the system and the model of the systems, e.g. the real-world thing that the system or a proof is supposed to apply to. Gödel's Incompleteness Theorem doesn't say anything about that. If you look at a proof as a set of formulae within a formal system - which is just a made-up language with a strict set of rules - you can say definitively that a proof's conclusion is true, at least to the extent that you check each step against the rules (which can sometimes be done by computer, improving the degree of certainty of the correctness check.)

But if you try to apply that proof to something in the real world, you have to perform a mapping between the formal system and the real world. There is room for mistakes here, and if you make a mistake, then a perfectly valid proof can result in an entirely wrong conclusion about the real world. However, this has nothing to do with Gödel. It's similar to the whole thing about lying with statistics: you can have perfectly correct statistics, but if you apply them incorrectly, the result is meaningless & therefore wrong.

However, this isn't a particularly serious problem, as it happens. It's possible to quite rigorously map a logical proof onto the real world. At some level, a certain amount of intuition is involved, but our entire philosophy of anything is based on such intuitional mappings. For example, we routinely map our abstract concept of number onto quantities of objects in the real world. If you add an apple to a group of two other apples, you get a group of three apples. Do you consider this conclusion "maybe valid, maybe not", as you put it? The applications of logic to the real world are just as reliable as our application of integers to the real world.

I also believe you can't dig deep enough to explain everything.

for example, we know that the number of integers is less than the number of reals